One-to-one

Inverse Functions Inverse Functions One-to-one One-to-one Suppose fff ::: A. Suppose A . A !B. * B isis aaa function. B function. We function. We call fff One-to-one We call call One-to-one ifif every every distinct distinct pair of pair of objects objects in A isis assigned in A. A assigned to to aaa distinct to distinct pair distinct pair of pair of objects objects in B. In in B. In other other words, each words, each object object of of the target has the target target has at has at most at most one most one object one object from from the the domain domain assigned assigned to to it. it. There There isis aa way way of of phrasing phrasing the previous definition the previous previous definition in definition in aa more more mathematical mathematical language: fff isis One-to-one language: language: One-to-one ifif whenever whenever we we have we have two have two objects 2e AA with objects a,a,cc with aa =. ~ c, 6= c, we are c, we are guaranteed guaranteed thatthat fff(a).

There is a way of phrasing the previous definition in a more mathematical language: f is one-to-one if whenever we have two objects a,c ⇤ A with a ⌅= c, we are guaranteed that f(a) ⌅= f(c). Example. f : R ⇥ R where f(x) = x2 is not one-to-one because 3 ⌅= 3 and yet f(3) = f(3) since f(3) and f(3) both equal 9. Horizontal line test

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